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The paper "Passenger and Cell Phone Conversations in Simulated Driving" (Journal of Experimental Psychology: Applied [2008]: 392-400) describes an experiment that investigated if talking on a cell phone while driving is more distracting than talking with a passenger. Drivers were randomly assigned to one of two groups. The 40 drivers in the cell phone group talked on a cell phone while driving in a simulator. The 40 drivers in the passenger group talked with a passenger in the car while driving in the simulator. The drivers were instructed to exit the highway when they came to a rest stop. Of the drivers talking to a passenger, 21 noticed the rest stop and exited. For the drivers talking on a cell phone, 11 noticed the rest stop and exited. a. Use the given information to construct and interpret a \(95 \%\) confidence interval for the difference in the proportions of drivers who would exit at the rest stop. b. Does the interval from Part (a) support the conclusion that drivers using a cell phone are more likely to miss the exit than drivers talking with a passenger? Explain how you used the confidence interval to answer this question.

Step by step solution

01

Calculate the sample proportions

Calculate the sample proportion of successful exits for each group: For passenger group (P1): \(n_{1} = 40\), \(x_{1} = 21\) Sample proportion, \( \hat{p}_{1} = \frac{x_{1}}{n_1} = \frac{21}{40} = 0.525\) For cell phone group (P2): \(n_{2} = 40\), \(x_{2} = 11\) Sample proportion, \( \hat{p}_{2} = \frac{x_{2}}{n_2} = \frac{11}{40} = 0.275\)

02

Calculate the difference in sample proportions

Calculate the difference in sample proportions: \(\hat{p}_{d} = \hat{p}_{1} - \hat{p}_{2} = 0.525 - 0.275 = 0.25\)

03

Calculate the standard error of the differences

Calculate the standard error of the differences: \(\text{SE}_{d} = \sqrt{\frac{\hat{p}_{1}(1-\hat{p}_{1})}{n_{1}} + \frac{\hat{p}_{2}(1-\hat{p}_{2})}{n_{2}}} = \sqrt{\frac{0.525(1-0.525)}{40} + \frac{0.275(1-0.275)}{40}} \approx 0.120\)

04

Construct the 95% Confidence Interval

Construct the 95% confidence interval for the difference in proportions: The margin of error is given by the product of the standard error and the z-score corresponding to a confidence level of 95% (z=1.96). So, ME = \(1.96 \times \text{SE}_{d} = 1.96 \times 0.120 \approx 0.235\) The confidence interval is given by the difference in sample proportions, plus or minus the margin of error: Lower limit: \(\hat{p}_{d} - \text{ME} = 0.25 - 0.235 = 0.015\) Upper limit: \(\hat{p}_{d} + \text{ME} = 0.25 + 0.235 = 0.485\) Therefore, the 95% confidence interval for the difference in proportions is \((0.015, 0.485)\). #b. Evaluating the Confidence Interval# Given that the entire confidence interval lies above zero, this means that, with 95% confidence, drivers talking to passengers are more likely to exit the rest stop than drivers talking on the cell phone. Since the interval does not include zero, there is evidence to support that drivers using cell phones are more likely to miss the exit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference in Proportions

In statistics, the difference in proportions is crucial when comparing two groups to see how one behaves compared to the other. In our exercise, we are comparing two groups of drivers: those talking to a passenger and those on a cell phone. The difference in proportions here refers to how many drivers in each group successfully exited at the rest stop.
The passenger group had a sample proportion of 0.525, meaning 52.5% of those drivers exited the highway successfully.
On the other hand, the cell phone group had a sample proportion of 0.275, indicating only 27.5% of drivers in this group succeeded in exiting.
The calculated difference (0.525 - 0.275) of 0.25 tells us there is a 25% greater probability that a driver talking with a passenger will exit compared to a driver talking on a cell phone.

Standard Error

The standard error (SE) measures how much we expect the sample estimates to vary from the true population parameters. It gives an idea of the accuracy of our sample's estimations.
In this exercise, the standard error for the difference in proportions was calculated to be approximately 0.120. This means our sample proportions' difference of 0.25 has some uncertainty because sampling is involved.

• A smaller standard error indicates more reliable results, as it means our sample result is closer to the actual population result.
• Larger standard errors suggest more variability and less confidence in the sample truly representing the population.

Understanding standard errors is crucial since it helps in forming confidence intervals which give us a range to express the statistical significance of results.

95% Confidence Level

When constructing a confidence interval, the 95% confidence level means that if we were to take 100 different samples and compute a confidence interval for each sample, about 95 of those intervals would contain the true population parameter.
This confidence level helps convey the reliability of our estimate. It doesn't guarantee accuracy but ensures a high probability that our interval truly includes the real difference.

This concept is essential in giving us a statistical assurance. In our exercise, the confidence interval for the difference in proportions was (0.015, 0.485). This interval being wholly above zero implies strong evidence that drivers conversing with passengers exit more than those using cell phones.

Experimental Design

Experimental design refers to the planned setup of an experiment, intending to investigate causal relationships or test hypotheses systematically.
In the original study, drivers were randomly assigned to either communicate with a passenger or on a cell phone while trying to exit a simulated highway. Here's why this design is effective:

• **Random Assignment:** Ensures each group is comparable and differences observed are due to the intervention (talking mode) rather than confounding factors.
• **Controlled Environment:** The driving simulator keeps conditions consistent across all participants, minimizing environmental variability.
• **Clear Measurement:** Exiting the highway serves as an objective indicator of distraction, offering clear data for analysis.

This careful experimental design strengthens the validity of the conclusions drawn, showing that cell phone conversations are more distracting than passenger conversations.